| L(s) = 1 | − 5-s − 2·9-s + 2·11-s + 2·19-s + 2·45-s − 49-s − 2·55-s + 2·61-s + 3·81-s − 2·95-s − 4·99-s − 4·101-s + 121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 4·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 5-s − 2·9-s + 2·11-s + 2·19-s + 2·45-s − 49-s − 2·55-s + 2·61-s + 3·81-s − 2·95-s − 4·99-s − 4·101-s + 121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 4·171-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5588885222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5588885222\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.73940428663520134005828723224, −11.39448347991777191910497595919, −11.21607423167003868914747075071, −10.63395831016736958726213973443, −9.680940375097192819510039946259, −9.564525582866044905900922091310, −9.043479528725945405742207084892, −8.559554467746743853534006034564, −8.101267339385240330948258433041, −7.79803134088040807684303816902, −6.94388774149781126870978416758, −6.77388089486647289399419825958, −5.99016608629263259011110941332, −5.57261185600090607275773745635, −5.04850928271146860858573829666, −4.23134329894390238626177522301, −3.60365775300727906009456452297, −3.31492373852692772330478789224, −2.51680543350416782780194495462, −1.22496783002337532100487424725,
1.22496783002337532100487424725, 2.51680543350416782780194495462, 3.31492373852692772330478789224, 3.60365775300727906009456452297, 4.23134329894390238626177522301, 5.04850928271146860858573829666, 5.57261185600090607275773745635, 5.99016608629263259011110941332, 6.77388089486647289399419825958, 6.94388774149781126870978416758, 7.79803134088040807684303816902, 8.101267339385240330948258433041, 8.559554467746743853534006034564, 9.043479528725945405742207084892, 9.564525582866044905900922091310, 9.680940375097192819510039946259, 10.63395831016736958726213973443, 11.21607423167003868914747075071, 11.39448347991777191910497595919, 11.73940428663520134005828723224
